1,2

Known lower bounds for the next terms are a(8)>=440943507851753, a(9)>=3.46254605664*10^17 and a(10)>=3.56944784623*10^20, found by Hermann Jurksch (jurksch(AT)elektron-bbs.de). The corresponding matrices are provided in the program given at the Pfoertner link. - Hugo Pfoertner (hugo(AT)pfoertner.org), Jan 22 2008

Hugo Pfoertner, Maximal determinant of matrix with elements 1..n. FORTRAN program.

Eric Weisstein's World of Mathematics, Square Matrix

The following 3 X 3 matrix is one of 36 whose determinant is 412 (there are also 36 whose determinant is -412):

9 3 5

4 8 1

2 6 7

Results from a specially-adapted hill-climbing algorithm strongly suggest that a(5) = 6839492. a(6) is at least 1862125166. Heuristically, a(n) is approximately 0.44 * n^(2.06*n), suggesting that a(7) is close to 6.8 * 10^11. - Tim Paulden (timmy(AT)cantab.net), Sep 21 2003

a(5) confirmed by Robert Israel (israel(at) math.ubc.ca) and Hugo Pfoertner. A corresponding matrix is ((25 15 9 11 4) (7 24 14 3 17) (6 12 23 20 5) (10 13 2 22 19) (16 1 18 8 21) ) - Hugo Pfoertner (hugo(AT) pfoertner.org), Sep 23 2003

a(6) found with FORTRAN program given at Pfoertner link. A corresponding matrix is ((36 24 21 17 5 8) ( 3 35 25 15 23 11) (13 7 34 16 10 31) (14 22 2 33 12 28) (20 4 19 29 32 6) (26 18 9 1 30 27) ) - Hugo Pfoertner (hugo(AT) pfoertner.org), Sep 23 2003

a(7) is the determinant of the matrix ((46 42 15 2 27 24 18) (9 48 36 30 7 14 31) (39 11 44 34 13 29 5) (26 22 17 41 47 1 21) (20 8 40 6 33 23 45) (4 28 19 25 38 49 12) (32 16 3 37 10 35 43)). Although no proof for the optimality of a(7) is available, the results of an extensive computational search make the existence of a better solution extremely unlikely. A total of approximately 15 CPU years on SGI Origin 3000 and of 3.8 CPU years on SGI Altix 3000 computers was used for this result.

Needs["DiscreteMath`Combinatorica`"]; n=3; n2=n^2; dMax=0; mMax={}; p=Range[n2]; Do[m=Partition[p, n]; d=Det[m]; If[d>dMax, dMax=d; mMax=m]; p=NextPermutation[p], {k, n2!}]; {dMax, mMax} (from T. D. Noe)

Cf. A088214, A088215, A088216, A088217, A088237.

Sequence in context: A162677 A041767 A130557 this_sequence A126154 A001327 A159533

Adjacent sequences: A084997 A084998 A084999 this_sequence A085001 A085002 A085003

nonn,nice,hard

Robert G. Wilson (rgwv(AT)rgwv.com), Jun 16 2003

a(4) from Marsac Laurent (jko(AT)rox0r.net), Sep 15 2003

a(6) from Hugo Pfoertner (hugo(AT)pfoertner.org), Sep 23 2003

Entry edited by N. J. A. Sloane (njas(AT)research.att.com), Nov 22 2006, to remove some erroneous entries. Further edits Nov 25 2006.

a(7) from Hugo Pfoertner (hugo(AT)pfoertner.org), Jan 22 2008